Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
2000 | 143 | 2 | 175-197
Tytuł artykułu

Selfsimilar profiles in large time asymptotics of solutions to damped wave equations

Treść / Zawartość
Warianty tytułu
Języki publikacji
Large time behavior of solutions to the generalized damped wave equation $u_{tt} +A u_t +ν B u+F(x,t,u,u_t,∇ u) = 0$ for $(x,t)∈ ℝ^n × [0,∞)$ is studied. First, we consider the linear nonhomogeneous equation, i.e. with F = F(x,t) independent of u. We impose conditions on the operators A and B, on F, as well as on the initial data which lead to the selfsimilar large time asymptotics of solutions. Next, this abstract result is applied to the equation where $Au_t = u_t$, $Bu = -Δu$, and the nonlinear term is either $|u_t|^{q-1}u_t$ or $|u|^{α-1}u$. In this case, the asymptotic profile of solutions is given by a multiple of the Gauss-Weierstrass kernel. Our method of proof does not require the smallness assumption on the initial conditions.
Opis fizyczny
  • Institute of Mathematics, Wrocław University, Pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland
  • [1] P. Biler, Partition of energy in strongly damped generalized wave equations, Math. Methods Appl. Sci. 12 (1990), 95-103.
  • [2] P. Biler, Time decay of solutions of semilinear strongly damped generalized wave equations, ibid. 14 (1991), 427-443.
  • [3] P. Biler, G. Karch and W. A. Woyczy/nski, Asymptotics for multifractal conservation laws, Studia Math. 135 (1999), 231-252.
  • [4] P. Biler, G. Karch and W. A. Woyczy/nski, Critical nonlinearity exponent and self-similar asymptotics for Lévy conservation laws, preprint, 1999, 29 pp.,
  • [5] D. B. Dix, The dissipation of nonlinear dispersive waves: The case of asymptotically weak nonlinearity, Comm. Partial Differential Equations 17 (1992), 1665-1693.
  • [6] J. Duoandikoetxea and E. Zuazua, Moments, masses de Dirac et décomposition de fonctions, C. R. Acad. Sci. Paris Sér. I Math. 315 (1992), 693-698.
  • [7] J. Dziuba/nski, Schwartz spaces associated with some non-differential convolution operators on homogeneous groups, Colloq. Math. 63 (1992), 153-161.
  • [8] M. Escobedo and E. Zuazua, Large time behavior for convection-diffusion equations in $ℝ^N$, J. Funct. Anal. 100 (1991), 119-161.
  • [9] T. Gallay and G. Raugel, Stability of travelling waves for a damped hyperbolic equation, Z. Angew. Math. Phys. 48 (1997), 451-479.
  • [10] T. Gallay and G. Raugel, Scaling variables and asymptotic expansions in damped wave equations, J. Differential Equations 150 (1998), 42-97.
  • [11] L. Hsiao and T.-P. Liu, Convergence to nonlinear diffusion waves for solutions of a system of hyperbolic conservation laws with damping, Comm. Math. Phys. 143 (1992), 599-605.
  • [12] L. Hsiao and T.-P. Liu, Nonlinear diffusive phenomena of nonlinear hyperbolic systems, Chinese Ann. Math. Ser. B 14 (1993), 465-480.
  • [13] S. Igari and S. Kuratsubo, A sufficient condition for $L^p$-multipliers, Pacific J. Math. 38 (1971), 85-88.
  • [14] G. Karch, $L^p$-decay of solutions to dissipative-dispersive perturbations of conservation laws, Ann. Polon. Math. 67 (1997), 65-86.
  • [15] G. Karch, Asymptotic behaviour of solutions to some pseudoparabolic equations, Math. Methods Appl. Sci. 20 (1997), 271-289.
  • [16] G. Karch, Selfsimilar large time behavior of solutions to Korteweg-de Vries-Burgers equation, Nonlinear Anal. 35 (1999), 199-219.
  • [17] G. Karch, Large-time behaviour of solutions to non-linear wave equations: higher-order asymptotics, Math. Methods Appl. Sci. 22 (1999), 1671-1697.
  • [18] S. Kawashima, M. Nakao and K. Ono, On the decay property of solutions to the Cauchy problem of the semilinear wave equation with a dissipative term, J. Math. Soc. Japan 47 (1995), 617-653.
  • [19] T.-T. Li and Y. M. Chen, Global Classical Solutions for Nonlinear Evolution Equations, Pitman Monogr. Surveys Pure Appl. Math. 45, Longman Scientific & Technical, Harlow, 1992.
  • [20] A. Matsumura, On the asymptotic behavior of solutions of semi-linear wave equations, Publ. Res. Inst. Math. Sci. Kyoto Univ. 12 (1976), 169-189.
  • [21] K. Nishihara, Decay properties of solutions of some quasilinear hyperbolic equations with strong damping, Nonlinear Anal. 21 (1993), 17-21.
  • [22] K. Nishihara, Convergence rates to nonlinear diffusion waves for solutions of system of hyperbolic conservation laws with damping, J. Differential Equations 131 (1996), 171-188.
  • [23] K. Nishihara, Asymptotic behavior of solutions of quasilinear hyperbolic equations with linear damping, ibid. 137 (1997), 384-395.
  • [24] K. Nishihara and T. Yang, Boundary effect on asymptotic behaviour of solutions to the p-system with linear damping, ibid. 156 (1999), 439-458.
  • [25] K. Ono, The time decay to the Cauchy problem for semilinear dissipative wave equations, Adv. Math. Sci. Appl. 9 (1999), 243-262.
  • [26] R. Racke, Decay rates for solutions of damped systems and generalized Fourier transforms, J. Reine Angew. Math. 412 (1990), 1-19.
  • [27] J. T. Sandefur Jr., Existence and uniqueness of solutions of second order nonlinear differential equations, SIAM J. Math. Anal. 14 (1983), 477-487.
  • [28] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Math. Ser. 30, Princeton Univ. Press, Princeton, NJ, 1970.
  • [29] E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Math. Ser. 43, Princeton Univ. Press, Princeton, NJ, 1993.
  • [30] H. Yang and A. Milani, On the diffusion phenomenon of quasilinear hyperbolic waves, Chinese Ann. Math. Ser. B 21 (2000), 63-70.
Typ dokumentu
Identyfikator YADDA
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.